1.
60² + 30² = 4500²
Opposite corner = 67.08 ft.
//
2.
6² + 8² = 100²
The wire must be 10 ft.
//
3.
7² + 14² = 245²
The rope is 15.65 ft.
//
4.
180² + 300² = 122400²
He ran 349.85 ft.
//
I'm not too sure about question 3.
The Pacific halibut fishery has been modeled by the differential equation dy dt = ky 1 − y K where y(t) is the biomass (the tota
l mass of the members of the population) in kilograms at time t (measured in years), the carrying capacity is estimated to be K = 9×107 kg, and k = 0.75 per year. (a) If y(0) = 2×107 kg, find the biomass a year later. (Round your answer to two decimal places.) ×107 kg (b) How long will it take for the biomass to reach 4×107? (Round your answer to two decimal places.) years
1 answer:
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Answer:
Part A)
Chorus:
Band:
Part B)
After 9 years:
The chorus will have about 41 people.
And the band will have 48 people.
Part C)
About approximately 11 years.
Step-by-step explanation:
We are given that there are 15 people in the chorus. Each year, number of people in the chorus increases by 12%. So, the chorus increases exponentially.
There are 30 people in the band. Each year, 2 new people join the band. So, the band increases linearly.
Part A)
Since after each year, the number of people in the chorus increases by 12%, the new population will be 112% or 1.12 of the previous population.
So, using the standard form for exponential growth:
Where <em>a</em> is the initial population and <em>r</em> is the rate of change.
We will substitute 15 for <em>a </em>and 1.12 for <em>r</em>. Hence:
This represents the number of people in the chorus after <em>t</em> years.
We are given that 2 new people join the band each year. So, it increases linearly.
Since there are already 30 people in the band, our initial point or y-intercept is 30.
And since 2 new people join every year, our slope is 2. Then by the slope-intercept form:
And by substitution:
This represents the number of people in the band after <em>t</em> years.
Part B)
We want to find the number of people in the chorus and the band after 9 years.
Using the chorus function, we see that:
There will be approximately 41 people in the chorus after 9 years.
And using the band function, we see that:
There will be 48 people in the band after 9 years.
Part C)
We want to determine after approximately how many years will the number of people in the chorus and band be equivalent. Hence, we will set the two functions equal to each other and solve for <em>t</em>. So:
Unfortunately, it is impossible to solve for <em>t</em> using normal analytic methods. Hence, we can graph them. Recall that graphically, our equation is the same as saying at what point will our two functions intersect.
Referring to the graph below, we can see that the point of intersection is at approximately (10.95, 51.91).
Hence, after approximately 11 years, both the chorus and the band will have approximately 52 people.
